MNRAS 430, 450–458 (2013)
doi:10.1093/mnras/sts642
Galaxy rotation curves in the Grumiller’s modified gravity
Hai-Nan Lin,1‹ Ming-Hua Li,1 Xin Li1,2 and Zhe Chang1,2
1 Institute
of High Energy Physics, Chinese Academy of Sciences, 100049 Beijing, China
Physics Center for Science Facilities, Chinese Academy of Sciences, 100049 Beijing, China
2 Theoretical
Accepted 2012 December 17. Received 2012 December 15; in original form 2012 September 19
ABSTRACT
The effective potential in the Grumiller’s modified gravity includes the Newtonian potential
and a Rindler term. The fitting to the rotation curve data of eight galaxies suggests a universal
Rindler acceleration a ≈ 0.30 × 10−10 m s−2 . We do a two-parameter fit first, with the massto-light ratio (ϒ ∗ ) and the Rindler acceleration (a) as free parameters. It is found that the
data of six out of the eight galaxies fit well with the prediction of theory in the range 0 r 40 kpc, although the theoretical curves show a tendency of arising beyond this range.
The Rindler accelerations of the six well-fitted galaxies have the same magnitude, with an
average value ā ≈ 0.30 × 10−10 m s−2 . Inspired by this fact, we then carry out a one-parameter
(ϒ ∗ ) fit to the six galaxies, with a fixed at ā, and find that the theory can still reproduce the
observation. The value of Rindler acceleration we get here is a quarter of that of Milgrom’s
modified Newtonian dynamics. For the rest two galaxies, NGC 5055 and DDO 154, the fitting
results are significantly improved if the photometric scale length (h) is included as another
free parameter.
Key words: galaxies: kinematics and dynamics – galaxies: photometry – galaxies: spiral.
1 I N T RO D U C T I O N
The observations on rotation velocities of stars around a galaxy
centre show significant discrepancies from Newtonian theory. According to Newtonian gravity, the rotation velocity is inversely proportional to the square root of distance from the centre of a galaxy.
However, the observed data often show an asymptotically flat rotation curve out to the furthest data points. Generally, there are three
different ways to solve this problem. The most direct assumption
is that there are a large number of non-luminous matters that have
not been measured yet (Begeman, Broeils & Sanders 1991; Persic,
Salucci & Stel 1996; Chemin, de Blok & Mamon 2011). However, after decades of years heavy research, no direct evidences
of the existence of dark matter have been found. This inspires the
astronomers and physicists to search for other explanations of the
discrepancy between the Newtonian dynamical mass and the luminous mass. The most popular way is to modified the Newton
dynamics (MOND; Milgrom 1983a,b). With only one universal parameter, i.e. the critical acceleration a0 , MOND has achieved great
success in explaining the mass discrepancy problem (Sanders 1996;
Sanders & Verheijen 1998).
Besides MOND, it is also possible to modify the Newtonian
gravity (MOG). According to MOG, Newtonian gravity is invalid at galaxy scales. Such theories include non-symmetric gravity
theory (NGT; Moffat 1995), metric-skew-tensor gravity (MSTG;
Moffat 2005; Brownstein & Moffat 2006), scalar-tensor-vector
gravity (STVG; Moffat 2006), Carmelian general relativity (CGR;
E-mail: [email protected]
Carmeli 2000), Horava–Lifshitz gravity theory (H–L theory;
Horava 2009a,b,c; Cardone et al. 2010, 2012), etc. All of them
can in a large degree explain the observed rotation curves.
Two years ago, Grumiller (2010) proposed an effective model
for gravity at large distance, and obtained an effective potential in
leading order consists of a Newtonian potential (∝1/r) and a Rindler
term (∝r). Grumiller & Preis (2011) showed that the Rindler force
is capable of explaining about 10 per cent of the Pioneer anomaly
(Anderson et al. 1998), and simultaneously ameliorates the shape
of galactic rotation curves. Most importantly, the Rindler term does
not spoil the Solar system precision tests.
In this paper, we investigate the galaxy rotation curves in the Grumiller’s modified gravity (Grumiller 2010). The rest of the paper is
organized as follows. In Section 2, we review briefly the Grumiller’s
modified gravity theory. In Section 3, we do a best-fitting procedure
to constrain the parameters. The two-parameter fit shows that six
out of eight galaxies have approximately the same Rindler acceleration ā ≈ 0.30 × 10−10 m s−2 . Then we carry out a one-parameter
fit, with a fixed at ā. It is found that the theory can still reproduce
the observation. For the rest two galaxies, the fitting results are significantly improved if the photometric scale length is included as
a free parameter. Finally, discussions and conclusions are given in
Section 4
2 G RU M I L L E R ’ S M O D I F I E D G R AV I T Y
In Grumiller’s modified gravity theory (Grumiller 2010), the space–
time is described by a spherically symmetric metric:
ds 2 = gαβ dx α dx β + 2 (dθ 2 + sin2 θ dφ 2 ),
(1)
C 2013 The Authors
Published by Oxford University Press on behalf of the Royal Astronomical Society
Rotation curves in the Grumiller’s gravity
where the two-dimensional metric gαβ (xγ ) and the surface radius
(xγ ) depend only on the coordinates xγ = {t, r}. The Einstein–
Hilbert action and the matter action are given by
√
(2)
SEH = − d4 x −gR
and
Sm = −
√
d4 x −gLm ,
(3)
where R is the Ricci scalar and Lm is the Lagrangian of the matter
part.
Generally, a four-dimensional spherical space–time manifold M
can be decomposed to the direct product of a two-dimensional radial submanifold L and a two-dimensional angular submanifold S,
M = L ⊗ S. The ‘spherical reduction’ process (Grumiller 2001)
simplifies the four-dimensional Einstein–Hilbert action to a specific two-dimensional dilation gravity model (see Appendix A for
details):
√
(4)
SEH = − d2 x −g[f ()R + 2(∂)2 − 2].
For the same reason, the matter action can be reduced to a twodimensional one,
√
Sm = − d2 x −g2 Lm .
(5)
Note that the metric gμν and the Ricci scalar R in equations (4) and
(5) belong to the two-dimensional submanifold L. For simplicity,
we will not explicitly distinguish the symbols on M and L here and
after. Since we are only interested in the vacuum solution, we will
not discuss the matter part bellow.
If we allow for IR modification to the model but require it to
be power-counting renormalizable and non-singularity of curvature
for large , the action equation (4) can be generalized to the form
(Grumiller 2010)
√
(6)
S = − d2 x −g[f ()R + 2(∂)2 − 2V ()],
where
V () ≡ 32 − 4a − 1.
(7)
Here, and a are two constants. We will show later that is related
to the cosmological constant, and a corresponds to the Rindler
acceleration.
Variation of the action in equation (6) gives the equations of
motion (EOM) (Grumiller 2010):
,
R = 2 g αβ ∇α ∂β + 6 − 4a
(8)
α
2(∇μ ∂ν − gμν ∇ ∂α ) − gμν (∂)2 = gμν V (),
where the first equality is obtained from varying the action with
respect to the scalar field , and the second equality is obtained
from varying the action with respect to the two-dimensional metric
gμν . ‘∇ μ ’ represents covariant derivatives.
The general solution of equation (8) can be obtained using the
gauge theoretic formulation given by Cangemi & Jackiw (1992). In
the Schwarzschild gauge where the metric is diagonal, we can find
the solution of the above EOM (Grumiller 2010):
2
,
gαβ dx α dx β = −K 2 dt 2 + dr
K2
(9)
= r,
451
where
2M
(10)
− r 2 + 2ar.
r
If a = = 0, equation (9) reduces to the Schwarzschild solution
in general relativity. While if M = = 0, equation (9) reduces to
the two-dimensional Rindler metric (Wald 1984). Since the twodimensional submanifold L is embedded into the four-dimensional
manifold M, the solution equations (9) and (10) induces a fourdimensional metric on M,
K2 ≡ 1 −
gμν = diag(−K 2 , 1/K 2 , r 2 , r 2 sin2 θ ).
(11)
A straightforward calculation gives the Ricci scalar of the fourdimensional manifold (see Appendix B for details):
R=−
2
2K 2 + 2 (−2KK − K 2 2 − K 2 + 1),
(12)
where the primes denote derivatives with respect to the radial distance r. Substituting equations (9) and (10) into equation (12), we
obtain
8a
R = 6 −
.
(13)
r
Consider a point particle with energy E and angular momentum
moving along a geodesic in the plane θ = π/2 in the background
of the metric equation (11). The conservation of energy and angular
momentum gives (Wald 1984)
2 dt
K dτ = E,
(14)
= .
r 2 dϕ
dτ
The normalization of four-velocity (gμν uμ uν = −1) gives another
equality:
2
2
2
dt
dr
dϕ
1
2
2
+ 2
+r
= −1.
(15)
−K
dτ
K
dτ
dτ
Using equation (14), equation (15) can be transformed into the
following form:
1 dr 2
1
= E 2 − V eff ,
(16)
2 dτ
2
where
V eff ≡
K2
2
1+
2
r2
2
M2
r 2
M
+ 2 − 3 −
r
2r
r
2
2
+ ar 1 + 2 + · · ·
r
=−
(17)
is the effective potential. We have dropped the constant terms on
the right-hand side (rhs) of equation (17).
The first two terms on the rhs of equation (17) are the classical
Newtonian potential and the centrifugal barrier, respectively. The
third term is the general relativity correction. The fourth term is the
cosmological constant term. The last term, which is proportional
to the Rindler acceleration a, is peculiar to Grumiller’s modified
gravity model. Since the Rindler acceleration term
2
RA
(18)
V = ar 1 + 2
r
depends on the angular momentum, this may imply that the modified
gravity depends on the orbital eccentricities of stars, but just their
positions. However, the second term on the rhs of equation (18) is
452
H.-N. Lin et al.
much small than the first term. In the International System of Units
(SI), equation (18) takes the form
2
(vr)2
v2
RA
V = ar 1 + 2 2 = ar 1 + 2 2 = ar 1 + 2 ,
c r
c r
c
(19)
where v is the rotational velocity of a star moving around the centre
of the galaxy and c is the speed of light. For most galaxies, v is of
the order of magnitude about 102 km s−1 and we have v 2 /c2 1.
Thus, the second term on the rhs of equation (18) can be neglected
compared to the first term.
In the case of vanishing cosmological constant and vanishing angular momentum (i.e. = = 0), the effective force corresponding
to equation (17) is given as
eff ∂V
M
= − 2 − a.
(20)
F eff = −
∂r r
Assuming that the mass-to-light ratio (ϒ ∗ ) is a constant and the disc
is infinitely thin, the mass profile of the stellar disc can be written
as
(r) = (0) e−(r/h) ,
(25)
where (0) = ϒ ∗ I(0) is the central surface mass density. The total
mass of the stellar disc is given by
∞
2πr(r) dr = 2π(0)h2 .
(26)
M=
0
The mass-to-light ratio is given as
ϒ∗ =
(0)
M
=
.
L
I (0)
(27)
The total luminosity L of a galaxy is related to its absolute magnitude
M as
==0
M − M = −2.5 log10
L
,
L
(28)
Thus, the rotation velocity of a testing particle moving in the potential of a point particle of mass M reads
M
+ ar = vN2 + ar,
(21)
v(r) =
r
√
where vN ≡ M/r is the rotation velocity derived from Newtonian
dynamics. This velocity has the asymptotic behaviour:
vN (r)
r → 0,
v(r) = √
(22)
ar
r → ∞.
where M and L are the absolute magnitude and total luminosity
of the sun, respectively.
The rotation velocity contributed by the exponential disc can be
given as (Freeman 1970)
M
v∗ (r) =
γ (r),
(29)
r
At small distances, the rotation velocity reduces to the Newtonian
case. The Rindler force dominates only at sufficiently large distances, such as the galaxy scales.
γ (r) ≡
3 B E S T- F I T T I N G P RO C E D U R E
To calculate the theoretical rotation velocity, we should know the
mass distribution of a galaxy. Generally, the mass in a galaxy contains two components: gas (mainly HI and He) and stellar disc.
Some early-type galaxies may also contain a bulge in the centre.
However, the mass of the bulge is often much smaller than that of
the stellar disc, and we will neglect it for simplicity. Since The HI
Nearby Galaxy Survey (THINGS)1 brought an unprecedented level
of precision to the measurement of the rotation curves of galaxies,
we choose the THINGS galaxies as the samples. The profile of
neutral hydrogen (HI) is read directly from the THINGS data cube
(robust weighted moment-0) using the task ELLINT of Groningen
Image Processing System (GIPSY)2 . Assuming that HI locates in an
infinitely thin disc, we calculate the rotation velocity contributed by
HI (v HI ) directly using the task ROTMOD of GIPSY.
The profile of stellar disc is derived from the photometric data.
The surface brightness of the stellar disc can be fitted well by an
exponential law (de Vaucouleurs 1959)
I (r) = I (0) e−(r/h) ,
(23)
where h is the scale length and I(0) is the surface brightness at the
centre of galaxy. The integration of equation (23) gives the total
luminosity
L = 2πI (0)h2 .
1
(24)
http://www.mpia-hd.mpg.de/THINGS/Data.html (Walter et al., 2008)
http://www.astro.rug.nl/~gipsy/, the data base is maintained by Hans Terlouw.
2
where
r r r r3 r K
−
I
K
.
I
0
0
1
1
2h3
2h
2h
2h
2h
(30)
Here, In and Kn (n = 0, 1) are the nth order modified Bessel functions
of the first and second kind, respectively.
The Newtonian velocity due to the combined contributions of gas
and stellar disc is given by
4 2
v + v∗2 ,
(31)
vN =
3 HI
where the factor 4/3 comes from the contribution of both helium
(He) and neutral hydrogen (HI). Here we assume that the mass ratio
of He and HI is MHe /MHI = 1/3. Any other gases are negligible
compared to HI and He. Combining equations (21) and (31), we get
the theoretical rotation velocity in the Grumiller’s modified gravity:
4 2
v (r) + v∗2 (r) + ar.
(32)
v(r) =
3 HI
In principle, equation (32) is only valid to spherically symmetric
systems. Since the potential in equation (17) is non-linear to mass,
superposition principle as well as Gauss theorem no longer holds,
and we cannot do an integration over the volume of mass as is
done in the case of Newtonian theory to obtain the potential of
any mass distribution. Any modified gravity is confronted with this
problem. We adopt an approximation as is done in MOND and
other modified gravities, by simply extrapolating equation (32) to
any mass distribution, at least to the axisymmetric cases.
Now we can use equation (32) to fit the observed rotation curves.
The free parameters are 0 (or ϒ ∗ ) and a. The sample galaxies
and their properties are listed in Table 1. The rotation velocity data
are read directly from the THINGS data cube (robust weighted
moment-1) using the task ROTCUR of GIPSY, with position angle (PA),
inclinations angle (INCL), Vsys and galaxy centre (RA and Dec.)
Rotation curves in the Grumiller’s gravity
453
Table 1. Properties of the sample galaxies. Column (1): galaxy names. Columns (2) and (3): galaxy centres in J2000.0 from Walter et al. (2008). Columns
(4) and (5): inclinations and position angles from Walter et al. (2008). Column (6): systematic velocities from de Blok et al. (2008). Column (7): distances
from Walter et al. (2008). Column (8): the scale lengths of optical disc from Begeman et al. (1991) and Flores (1993), but corrected with the new distances in
column (7). Column (9): mass of HI from Walter et al. (2008). Column (10): apparent B-band magnitudes from Walter et al. (2008). Column (11): absolute
B-band magnitudes from Walter et al. (2008). Column (12): luminosity calculated from column (11) using equation (28).
(1)
Names
NGC 2403
NGC 2841
NGC 2903
NGC 3198
NGC 3521
NGC 5055
NGC 7331
DDO 154
(2)
RA
(h m s )
(3)
Dec.
(◦ )
(4)
INCL
(◦ )
(5)
PA
(◦ )
(6)
Vsys
(km s−1 )
(7)
D
(Mpc)
(8)
h
(kpc)
(9)
MHI
(108 M )
(10)
mB
(mag)
(11)
MB
(mag)
(12)
L
(1010 L )
07 36 51.1
09 22 02.6
09 32 10.1
10 19 55.0
10 05 48.6
13 15 49.2
22 27 04.1
12 54 05.9
+65 36 03
+50 58 35
+21 30 04
+45 32 59
−00 02 09
+42 01 45
+34 24 57
+27 09 10
63
74
65
72
73
59
76
66
124
153
204
215
340
102
168
230
132.8
633.7
555.6
660.7
803.5
496.8
818.3
375.9
3.2
14.1
8.9
13.8
10.7
10.1
14.7
4.3
2.05
3.55
2.81
3.88
2.86
5.00
4.48
0.54
25.8
85.8
43.5
101.7
80.2
91.0
91.3
3.58
8.11
9.54
8.82
9.95
9.21
8.90
9.17
13.94
−19.43
−21.21
−20.93
−20.75
−20.94
−21.12
−21.67
−14.23
0.920
4.742
3.664
3.105
3.698
4.365
7.244
0.008
fixed at the values given in Table 1. In order to constrain the parameters, we define the chi-square as
2
n
viobs − v th (ri )
,
(33)
χ2 =
σi2
i=1
where viobs is the observed rotation velocity, v th (ri ) is the theoretical
velocity at radius ri calculated from equation (32) and σ i is the
uncertainty of viobs . Then we employ the least-square method to
minimize equation (33). The best-fitting results are presented in
Fig. 1, and the values of parameters are listed in Table 2.
From Table 2, we can see that the Rindler acceleration a has the
same magnitude for all the eight galaxies except for NGC 5055 and
DDO 154, with an average value ā ≈ 0.30 × 10−10 m s−2 . Inspired
by this fact, we carry out a one-parameter fit procedure, with a
fixed at ā and (0) as the only free parameter. The best-fitting
results are shown in Fig. 2, and the values of parameters are listed
in Table 3. From Fig. 2, we find that the theory can still fits the
observations (except for NGC 3198), although the theoretical curves
show a tendency of arising beyond the data range. For the two
poorly fitted galaxies, NGC 5055 and DDO 154, we include the
disc scale length h as a free parameter and do a three-parameter fit.
The results are plotted in Fig. 3, and the best-fitting parameters are
listed in Table 4. It turns out that the fitting results are significantly
improved. However, the best-fitting scale lengths far deviate from
the photometric scale lengths. We do not know if it means that the
mass scale length differs from that of photometry.
4 DISCUSSION AND CONCLUSION
In this paper, we have done a best-fitting procedure to the rotation
curves of eight THINGS galaxies in the framework of Grumiller’s
modified gravity. We did a two-parameter fit first, with 0 and a as
free parameters. For the eight sample galaxies, six of them fit well
in the range 0 r 40 kpc. We found that the Rindler accelerations
of each galaxies share the same magnitude, with an average value
ā ≈ 0.30 × 10−10 m s−2 . The Rindler acceleration we got here is
a quarter of the critical acceleration of Milgrom’s MOND (a0 ≈
1.2 × 10−10 m s−2 ). Inspired by this fact, we assumed that a may be a
universal constant, and did another one-parameter fit to the six wellfitted galaxies, with a fixed at ā and ϒ ∗ as the only free parameter. It
turned out that the theoretical rotation curves can still reproduce the
observed data, with only one exception. The theoretical velocity of
NGC 3198 shows a tendency of sharply arising after ∼20 kpc. For
the rest two galaxies, NGC 5055 and DDO 154, if we include the disc
scale length as a free parameter, the fitting results are significantly
improved. Thus, the Rindler acceleration of all the eight galaxies,
except for DDO154, can be unified to ā. However, the Rindler
acceleration of DDO154, which is a gas-rich dwarf galaxy, is one
order of magnitude smaller.
A question may arise on applying a spherical symmetry built-in
theory to the disc galaxies. In fact, spherical solutions are often
obtained for theoretical studies of a gravity model or theory. The
solution is then extrapolated to other asymmetric systems under
certain conditions. A good example is Bekenstein’s TeVeS theory.
In Bekenstein (2004), the famous MOND was obtained in a spherically symmetric situation. He also pointed out that for a generically
asymmetric system, a curl term ∇ × h arises in the solution of the
Poisson’s equation of TeVeS (h is some regular vector field). This
term falls off rapidly at large distances thus is negligible well outside the system. The theory applies well outside any non-spherical
galaxy just as it applies anywhere inside a spherical one. In his
paper, Bekenstein showed that it is reasonable to use a spherical
symmetry built-in theory to study a non-spherical matter system.
The above conclusion is not valid under some circumstances,
such as the interior and the near-exterior of a non-spherical galaxy.
There also exist qualitative differences: stars cannot escape in a
spherical MOND potential, but are able to escape when a nonspherical external field is included (Wu et al. 2008). In these situations, one should solve the Poisson’s equation for an asymmetric
matter distribution by numerical methods like Milgrom (1986). For
MOND, this method has been used to give fruitful results in regard
to low surface brightness disc galaxies, dwarf spheroidal galaxies
and the outer regions of spiral galaxies (de Blok & McGaugh 1998;
Llinares, Knebe & Zhao 2008; Swaters, Sanders & McGaugh 2010;
Angus 2012). Therefore, to apply Grumiller’s spherically symmetric model to calculate the rotation curves of non-spherical spiral
galaxies, one should follow Milgrom’s way to solve numerically
the Poisson’s equation of Grumiller’s theory for axisymmetric matter distributions, e.g. an exponential stellar disc ρ(r, z) (in cylindrical
coordinates r, θ, z). Like MOND, this procedure is computationally expensive. Another way was given by Swaters et al. (2010).
For MOND, they used the usual formula g M = g N /μ(g
√ M /a0 ) (a0
is the MOND acceleration parameter and μ(x) ≡ x/ 1 + x 2 ) to
454
H.-N. Lin et al.
Figure 1. Two-parameter fit (black curves) to the rotation curves of sample galaxies. The x-axis is the distance in kpc, and the y-axis is the rotation velocity
in km s−1 . The red curves are the contribution of gas (HI and He). The green curves are the contribution of gas and stellar disc in Newtonian dynamics.
Table 2. The two-parameter best fit. The free parameters are 0 and a. Mdisc is the mass
of the disc calculated from equation (26). M/L is the mass-to-light ratio calculated from
equation (27). χ 2 /n is the reduced chi-square. The numbers after ‘±’ are the 1σ errors of the
corresponding parameters.
NGC 2403
NGC 2841
NGC 2903
NGC 3198
NGC 3521
NGC 5055
NGC 7331
DDO 154
0
(M pc−2 )
a
(10−10 m s−2 )
Mdisc
(1010 M )
M/L
(M /L )
χ 2 /n
450.75 ± 9.73
2626.12 ± 269.17
1190.88 ± 24.66
421.25 ± 9.75
1239.17 ± 40.85
900.27 ± 14.72
1049.41 ± 53.30
92.76 ± 46.23
0.301 ± 0.006
0.396 ± 0.056
0.316 ± 0.010
0.160 ± 0.005
0.296 ± 0.014
0.053 ± 0.013
0.299 ± 0.021
0.060 ± 0.010
1.19 ± 0.03
20.79 ± 2.13
5.91 ± 0.12
3.98 ± 0.09
6.37 ± 0.21
14.14 ± 0.23
13.23 ± 0.67
0.02 ± 0.01
1.29 ± 0.03
4.39 ± 0.50
1.61 ± 0.03
1.28 ± 0.03
1.72 ± 0.06
3.24 ± 0.05
1.83 ± 0.09
2.12 ± 1.06
1.21
2.47
3.87
2.49
0.54
5.84
0.35
17.48
Rotation curves in the Grumiller’s gravity
455
Figure 2. One-parameter fit with a fixed at ā ≈ 0.30 × 10−10 m s−2 . The x-axis is the distance in kpc, and the y-axis is the rotation velocity in km s−1 . The
red curves are the contribution of gas (HI and He). The green curves are the contribution of gas and stellar disc in Newtonian dynamics.
obtain the MOND acceleration g M as a function of the Newtonian
acceleration g N , i.e.
g 2M = g 2N 1 + 1 + 4a02 /g 2N
2.
(34)
Using the above relation, the circular velocity v M (r) within the
MOND framework can be expressed as a function of a0 and the
2
(r) = vN2 (r)[ 12 (1 +
Newtonian baryonic contribution v N (r) as vM
√
1 + (2ra0 /vN2 (r))2 ]1/2 , where vN (r) ≡ GN M/r, GN is the
Newtonian gravity constant. The stellar contribution of v N (r) was
obtained by solving the Newtonian Poisson’s equation for an axisymmetric density distribution ρ(r, z) ∝ e−z/h (in cylindrical coordinates (r, θ, z), h is the scale height of the stellar disc). The
result was given analytically as equations (29) and (30) (Freeman
1970; Casertano 1983). According to Milgrom (1986), the differences between the results obtained by using this method and those
Table 3. The one-parameter best fit. The free parameter is 0 . The parameter a is fixed at the average value ā ≈ 0.30 × 10−10 m s−2 . Mdisc is the mass
of the disc calculated from equation (26). M/L is the mass-to-light ratio
calculated from equation (27). χ 2 /n is the reduced chi-square. The numbers
after ‘±’ are the 1σ errors of the corresponding parameters. We skip the
one-parameter fit for NGC 5055 and DDO 154.
NGC 2403
NGC 2841
NGC 2903
NGC 3198
NGC 3521
NGC 5055
NGC 7331
DDO 154
0
(M pc−2 )
Mdisc
(1010 M )
M/L
(M /L )
χ 2 /n
444.96 ± 10.78
2825.29 ± 93.10
1209.00 ± 34.15
248.29 ± 13.20
1222.09 ± 64.39
×
1037.32 ± 72.80
×
1.17 ± 0.03
22.37 ± 0.74
6.00 ± 0.17
2.35 ± 0.12
6.28 ± 0.33
×
13.08 ± 0.92
×
1.28 ± 0.03
4.72 ± 0.16
1.64 ± 0.05
0.76 ± 0.04
1.70 ± 0.09
×
1.81 ± 0.13
×
1.23
4.15
3.89
18.79
0.55
×
0.36
×
456
H.-N. Lin et al.
Figure 3. Three-parameter fit with 0 , a and h free. The x-axis is the distance in kpc, and the y-axis is the rotation velocity in km s−1 . The red curves are the
contribution of gas (HI and He). The green curves are the contribution of gas and stellar disc in Newtonian dynamics.
Table 4. The three-parameter best fit. The free parameters are 0 , a and h. Mdisc is the mass of the disc
calculated from equation (26). M/L is the mass-to-light ratio calculated from equation (27). χ 2 /n is the
reduced chi-square. The numbers after ‘±’ are the 1σ errors of the corresponding parameters.
0
(M pc−2 )
NGC 5055
DDO 154
1503.24 ± 217.05
47.86 ± 3.87
a
m s−2 )
(10−10
0.245 ± 0.042
0.027 ± 0.005
by numerically solving the Poisson’s equation of MOND are usually
much smaller than 5 per cent, although in some cases it may be
much larger (Angus, Famaey & Zhao 2006). The second method is
computationally more economical so that it has been widely used
to calculate the circular velocity for low surface brightness disc
galaxies, dwarf and other non-spherical galaxies (Gentile 2008;
Swaters et al. 2010).
Thus, considering all these mentioned above, it is reasonable
to apply Grumiller’s gravity model to non-spherical disc galaxies,
even though the theory was first contrived in spherical symmetry. All one has to do is to make the Newtonian approximation of
the theory and then solve the corresponding Poisson’s equation for
non-spherical density distributions, like the MOND and other modified gravity theories. In this paper, we took the second path in the
above discussions by substituting equations (29) and (30) into the
relation between the acceleration for Grumiller’s model g G and the
Newtonian acceleration g N , i.e.
g G = g N + a,
(35)
where a is called the Rindler acceleration. This relation has its
analog in MOND as equation (34). Thus, the rotational
velocity in
√
Grumiller’s modified gravity is given as vG (r) = GN M/r + ar =
vN2 + ar. It is simply the equation (32). We based our numerical
analysis on this equation.
In addition, a few comments should be given on the parameter
a. In our numerical study, we regarded the Rindler acceleration
as a constant for a certain galaxy. Grumiller (2010) argued that a
may vary from system to system. Furthermore, a is not necessary
to be a constant even in the same system: it may be a function
of r. For instance, a is one order of magnitude larger in the Sun–
Pioneer system than that of in the Galaxy–Sun system. In the Earth–
Satellite system, it may be much larger. Otherwise, the theory cannot
reconcile with all the experiments. Unfortunately, it is still unknown
what determines the scale of a. In this paper, however, we found that
h
(kpc)
Mdisc
(1010 M )
M/L
(M /L )
χ 2 /n
2.81 ± 0.32
2.45 ± 0.28
7.46 ± 2.78
0.18 ± 0.06
1.71 ± 0.64
22.56 ± 6.98
2.50
0.86
a may be a universal constant. It is surprising that the value of a we
obtained here is the same as the upper bound value constrained from
the Cassini spacecraft data (Grumiller & Preis 2011). Furthermore, a
fitting to the Burkert’s dark matter profile shows that within the scale
length r0 , the mean dark matter surface density is approximately a
constant for galaxies spanning a luminosity range of 14 mag. This
leads to a constant gravity acceleration at the radius r0 , i.e. g(r0 ) ≈
0.3 × 10−10 m s−2 (Donato et al. 2009; Gentile 2009), which is
in good agreement with our result. It is interesting that the same
acceleration is measured in so many systems. This motivates us to
investigate the modified gravity theories in which such a scale exists
naturally.
The most interesting feature of Grumiller’s gravity is that it predicts a rotation curve proportional to the square root of distance in
the large distance limits. However, the observed data just extend to
a few tens of kpc. Although the velocity law equation (32) follows
well to the experiment data in this range, it is too early to say that this
is a valid model at large distances. Future observations of rotation
velocity at large distances may provide further tests to the theory. It
is very likely that Grumiller’s gravity will fail in the large distance
limits if a is a constant. As was mentioned
√ by Grumiller (2010), a
may be a function of r. If we choose a = GMa0 /r, equation (21)
becomes
(36)
v 2 = vN2 + GM(r)a0 ,
where a0 ≈ 1.2 × 10−10 m s−2 is the critical acceleration in Milgrom’s MOND and M(r) is the mass surrounded by a sphere of radius r. Equation (36) has the same asymptotic behaviour as MOND,
with the Tully–Fisher relation holds. Li & Chang (2012) obtained
a similar formula with the same asymptotic behaviour as equation (36) based on the Finsler geometry. Li et al. (2012) showed that
equation (36) plus the dipole and quadrupole contributions can well
explain the mass discrepancy of Bullet Cluster 1E 0657−558. Since
the observed data show that most galaxies have asymptotically flat
Rotation curves in the Grumiller’s gravity
rotation curves, it may better describe the galaxy rotation curves if
a is inversely proportional to r than is a constant.
AC K N OW L E D G M E N T S
We are grateful to Y. G. Jiang and S. Wang for useful discussions.
This work has been funded in part by the National Natural Science
Fund of China under Grant No. 10875129 and No. 11075166.
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457
ity model. Generally, a four-dimensional spherical space–time
manifold M can be decomposed to the direct product of 2 twodimensional submanifolds, M = L ⊗ S, where the radial part
L depends only on the coordinates (t, r). At the same time, the
Ricci scalar on M induces the Ricci scalar on L. Given the fourdimensional spherical metric
ds 2 = gαβ dx α dx β + 2 (dθ 2 + sin2 θ dφ 2 ),
(A1)
where the Greek index runs in (t, r) and the two-dimensional metric
gαβ and the dilaton field depend only on (t, r), a straight forward
calculation shows that (Grumiller 2001)
R (M) = R (L) −
2
4
[1 + (∇)2 ] − (),
2
(A2)
where R(M) and R(L) are the Ricci scalar on M and L, respectively.
‘∇’ and ‘’ are the covariant derivative operator and the Laplacian
operator on L, respectively.
On the other hand, the determinant of the metric on M can be
written as
g (M) = g (L) 4 sin2 θ.
(A3)
Substituting equations (A2) and (A3) into the Einstein–Hilbert action, and integrating over the angles (θ , ϕ), we get
SEH = − d4 x −g (M) R
= −4π
d2 x
−g (L) 2 R (L) − 2 1 + (∇)2 − 4 .
(A4)
Integrating the last term by part and dropping the surface term, we
get
(A5)
d2 x −g (L) (−4) = 4 d2 x −g (L) (∇)2 .
Thus, equation (A4) becomes
SEH = −4π d2 x −g (L) 2 R (L) + 2(∇)2 − 2 .
(A6)
Similarly, for a spherically symmetric matter Lagrangian Lm =
Lm (t, r), the four-dimensional action can be reduced to a twodimensional one:
(A7)
Sm = − d4 x −g (M) Lm = −4π d2 x −g (L) 2 Lm .
APPENDIX B: RICCI SCALAR
For a given four-dimensional spherical metric
gμν = diag(−K 2 , 1/K 2 , 2 , 2 sin2 θ ),
(B1)
− r + 2ar and ≡ r, the non-vanishing
where K ≡ 1 −
components of the metric and its inverse are
2
g00 = −K 2 ,
2M
r
g11 =
2
1
,
K2
g22 = 2 ,
g33 = 2 sin2 θ,
(B2)
APPENDIX A: SPHERICAL REDUCTION
For a spherical metric, it is possible to reduce the four-dimensional
Einstein–Hilbert action to a specific two-dimensional dilaton grav-
g 00 = −
1
,
K2
g 11 = K 2 ,
g 22 =
1
,
2
g 33 =
1
. (B3)
2 sin2 θ
458
H.-N. Lin et al.
A straightforward calculation shows that the non-vanishing
Christoffel connections are given as
⎧ 0
0
01 = 10
= 12 g 00 g00,1 = K1 dK
,
⎪
dr
⎪
⎪
⎪
⎪
1
⎪
= − 12 g 11 g00,1 = −K 3 dK
,
⎪ 00
⎪
dr
⎪
⎪
⎪
1
⎪
⎪ 11
= 12 g 11 g11,1 = − K1 dK
,
⎪
dr
⎪
⎪
⎪
⎪
1
1
11
2
⎪
,
= − 2 g g22,1 = −K d
⎪
dr
⎪
⎨ 22
1 11
1
2 d
33 = − 2 g g33,1 = −K dr sin2 θ,
(B4)
⎪
⎪
⎪
⎪ 2 = 2 = 1 g 22 g22,1 = 1 d ,
⎪
⎪
12
21
2
dr
⎪
⎪
⎪
⎪
1 22
2
⎪
⎪
⎪ 33 = − 2 g g33,2 = − sin θ cos θ,
⎪
⎪
⎪
3
3
⎪
= 31
= 12 g 33 g33,1 = 1 d
,
13
⎪
dr
⎪
⎪
⎪
⎩ 3
1 33
3
23 = 32 = 2 g g33,2 = cot θ.
The non-vanishing components of the Ricci tensor are
⎧
3
⎪
R00 = −K 2 K 2 − K 3 K − 2K
K ,
⎪
⎪
⎪
⎪
⎪
⎨ R = − K − 2
− 2K − K 2 ,
11
K
K
K2
⎪
2 2
⎪
R22 = −2KK − K − K 2 + 1,
⎪
⎪
⎪
⎪
⎩
R33 = (−2KK − K 2 2 − K 2 + 1) sin2 θ.
(B5)
The primes in the above equations denote the differentiation with
respect to the radial distance r. Thus, the Ricci scalar is given as
R ≡ Rμν g μν =
−
2K 2 2
+ 2 (−2KK −K 2 2 − K 2 + 1).
(B6)
This paper has been typeset from a TEX/LATEX file prepared by the author.